Robust lyapunov controller for uncertain systems

ABSTRACT

Various examples of systems and methods are provided for Lyapunov control for uncertain systems. In one example, a system includes a process plant and a robust Lyapunov controller configured to control an input of the process plant. The robust Lyapunov controller includes an inner closed loop Lyapunov controller and an outer closed loop error stabilizer. In another example, a method includes monitoring a system output of a process plant; generating an estimated system control input based upon a defined output reference; generating a system control input using the estimated system control input and a compensation term; and adjusting the process plant based upon the system control input to force the system output to track the defined output reference. An inner closed loop Lyapunov controller can generate the estimated system control input and an outer closed loop error stabilizer can generate the system control input.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to, and the benefit of, co-pending U.S.provisional application entitled “ROBUST LYAPUNOV CONTROLLER FORUNCERTAIN SYSTEMS” having Ser. No. 62/205,631, filed Aug. 14, 2015,which is hereby incorporated by reference in its entirety.

BACKGROUND

Nowadays, the decrease in fossil resources along with the increase oftheir environmental impact is driving the interest of public and privatesectors in renewable energy to meet the world demand. Indeed, differentforms of renewable energy present promising alternative clean sources,as part of them, solar energy. The operation of any solar driven plantis constrained by the availability of the sunlight, which ischaracterized by its intermittency and unpredictable changes affectingthe energy production. Thereafter, in order to better exploit solarenergy, research investigations are gradually consolidated, from bothtechnological and control perspectives, in order to enhance theproduction efficiency. Solar thermal distributed concentratedtechnology, known for high efficiency with available thermal storagesolution, is one of the main researchers' interests.

SUMMARY

Embodiments of the present disclosure are related to Lyapunov controlfor uncertain systems such as, e.g., parabolic distributed collectors.

In one embodiment, among others, a system comprises a process plant anda robust Lyapunov controller configured to control an input of theprocess plant. The robust Lyapunov controller comprises an inner closedloop Lyapunov controller and an outer closed loop error stabilizer. Inone or more aspects of these embodiments, the process plant can be adistributed solar collector and the input of the process plant is aninlet fluid flow rate. The process plant can comprise a parabolic solarcollector.

In one or more aspects of these embodiments, the robust Lyapunovcontroller can comprise a nominal model of the process plant configuredto generate an estimated output based at least in part upon fixedworking conditions of the process plant. The nominal model can be aphysical distributed model of the process plant. The physicaldistributed model can be a bilinear model that approximates the processplant by a low order nonlinear set of ordinary differential equationsusing dynamical Gaussian interpolation.

In one or more aspects of these embodiments, the outer closed loop errorstabilizer can be configured to generate the input to force an output ofthe process plant to track a nominal output using a phenomenologicalrepresentation of the process plant. The inner closed loop Lyapunovcontroller can be configured to generate an estimated input based uponan estimated output generated by a nominal model of the process plantand an output reference, the estimated input provided to the outerclosed loop error stabilizer and the nominal model of the process plant.

In another embodiment, a method comprises monitoring an output of aprocess plant; generating an estimated control input based at least inpart upon a defined output reference; generating a control input basedat least in part upon the estimated control input and a compensationterm based upon a phenomenological model of the process plant; andadjusting operation of the process plant based upon the control input toforce the output of the process plant to track the defined outputreference. An inner closed loop Lyapunov controller can generate theestimated control input and an outer closed loop error stabilizer cangenerate the control input.

In one or more aspects of these embodiments, the control input can befurther based upon a difference between an estimated output and theoutput being monitored, where the estimated output is based upon anominal model of the process plant and the estimated control input. Thenominal model can be a physical distributed model of the process plant.The physical distributed model can be a bilinear model that approximatesthe process plant by a low order nonlinear set of ordinary differentialequations using dynamical Gaussian interpolation.

In one or more aspects of these embodiments, the control input can befurther based upon time varying weighting parameters provided by thenominal model. The compensation term can be determined based upon thecontrol input and a rate of change of the output. The process plant is adistributed solar collector and the input of the process plant is aninlet fluid flow rate.

Other systems, methods, features, and advantages of the presentdisclosure will be or become apparent to one with skill in the art uponexamination of the following drawings and detailed description. It isintended that all such additional systems, methods, features, andadvantages be included within this description, be within the scope ofthe present disclosure, and be protected by the accompanying claims. Inaddition, all optional and preferred features and modifications of thedescribed embodiments are usable in all aspects of the disclosure taughtherein. Furthermore, the individual features of the dependent claims, aswell as all optional and preferred features and modifications of thedescribed embodiments are combinable and interchangeable with oneanother.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the present disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIGS. 1A and 1B illustrate an example of a parabolic solar collector inaccordance with various embodiments of the present disclosure.

FIG. 2 includes a table of system parameters of a solar collector inaccordance with various embodiments of the present disclosure.

FIG. 3 is an example of radial basis functions distributed along a solarcollector in accordance with various embodiments of the presentdisclosure.

FIG. 4 is an example of a solar irradiance profile in accordance withvarious embodiments of the present disclosure.

FIG. 5 is an example of a fluid flow rate profile of a solar collectorin accordance with various embodiments of the present disclosure.

FIG. 6 includes examples of the temperature dynamics of the solarcollector in accordance with various embodiments of the presentdisclosure.

FIGS. 7 and 8 graphically illustrate approximation error of a bilinearmodel in accordance with various embodiments of the present disclosure.

FIG. 9 is a schematic diagram of an example of a robust Lyapunovcontroller in accordance with various embodiments of the presentdisclosure.

FIGS. 10A and 10B graphically illustrate examples of an inhomogeneousdistribution of varying parameters in accordance with variousembodiments of the present disclosure.

FIGS. 11A-11C, 12A-12C and 13A-13C illustrate examples of (a) solarirradiance, (b) reference tracking, and (c) control input for threetests of the robust controller of FIG. 9 in accordance with variousembodiments of the present disclosure.

FIG. 14 is a schematic block diagram that illustrates an example ofprocessing circuitry employed by the robust controller of FIG. 9 inaccordance with various embodiments of the present disclosure.

DETAILED DESCRIPTION

Disclosed herein are various embodiments of methods related to Lyapunovcontrol for uncertain systems such as, e.g., parabolic distributedcollectors. Reference will now be made in detail to the description ofthe embodiments as illustrated in the drawings, wherein like referencenumbers indicate like parts throughout the several views.

The parabolic distributed technology, in particular, is widely used tofeed solar thermal processes. The parabolic troughs are spatio-temporaldistributed systems working based on a heat transfer mechanism where theparabolic shaped mirrors concentrate the received solar irradiance to areceiver tube in order to heat the thermal carrier fluid flowing throughit. Moreover, considering that the solar collector is subject to variousrandom external disturbances, the system time varying dynamics are richenough to present difficulties for control design. In addition, theannual pluvial cycles, the humidity levels and the dust accumulationmodify considerably the reflectivity of the mirrors, adding challengesfor efficient control schemes. Therefore, the design of advanced controlstrategies able to reject unevenly distributed solar irradiance and tocope for the time delays related to transport phenomena is desired.

The control problem generally addressed in solar plants using parabolicshaped collectors aims at forcing the field outlet temperature to tracka set reference in order to manage the heat production. The objective isto design an efficient controller able to maintain the desired fluidoutlet temperature despite disturbances in irradiation and systemparameters, by tuning the inlet fluid flow rate as a control input. Todeal with this problem, the control design has been approached by meansof many different techniques (conventional feedback control, modelpredictive control, adaptive control, internal model control, fuzzycontrol, or more complex control strategies). Developed controltechniques can be applied to distributed concentrated collectors.Control techniques can really enhance the performance of solar thermalpower plants. Indeed, developing advanced control strategies for theparabolic troughs fields continues to be a challenge to control theory,especially that these plants are representative of a wider category ofthermal processes working based on a one dimensional heat conductionmechanism.

A well designed control system for solar plants should be able tominimize the problems related to the solar irradiation intermittency fora more efficient functioning. The estimation of the efficient value ofthe source term has been used as a solution to continuously update thecontroller in order to compensate for the external disturbances.Numerous control techniques have been combined with estimation oridentification methodologies such as the indirect adaptive control,model predictive control, adaptive fuzzy switching control techniques ormore advanced adaptive techniques. Moreover, to reduce the impact ofdisturbances, feedforward control can be efficient.

Measurable disturbances can be utilized to implement a feedforwardcontroller, which are based upon reliable measurements of the externalperturbations. In commercial solar systems not all variables andparameters are monitored by direct measurements, since some of them maybe technically difficult or expensive to measure. While reflectometersand pyheliometers offer local measurements, it is difficult to obtain anadequate estimation of metal tubes absorptance and other unknownfactors. Thereafter, for more efficient control of the system withoutadditional computational effort for the estimation of the source term,the design of a robust control law with respect to the varying systemparameters and external disturbances is a judicious choice. Theparabolic trough solar collector is a distributed nonlinear constrainedsystem. Several techniques presented in the literature are designedassuming the plant to be modeled as a black box lumped parameterssystem. Due to the wide operating range of solar collector plants, thelinear models are in general affected by a significant parametricuncertainty.

Consideration of a distributed physical model for the control design canimprove their performance. In addition to the distributed systemdynamics, the system is nonlinear with respect to the control input andcan be characterized by a randomly varying source term. Therefore,numerical schemes can be applied on the hyperbolic heat transportequation, while preserving the distributed nature of the dynamics, inorder to derive a set of ordinary differential equations (ODEs) to beused for system analysis and/or control design. To reproduce the systemdynamics accurately, a high number of knots in the computational gridmay be needed which could increase the computational effort andcomplicate the control design. In this respect, an approximate bilinearmodel can be utilized. The hyperbolic distributed model of the parabolictroughs can be approximated by a low order nonlinear set of ODEs usingdynamical Gaussian interpolation. The resulting model takes the form ofa reduced order bilinear state space representation, which can be usedas a basis for the robust control design.

In the present disclosure, a robust nonlinear output feedback control isdeveloped for reference tracking of the desired temperature level. Thedesign combines Lyapunov state feedback with a phenomenologicalrepresentation of the plant to force the measured outlet temperature totrack the desired reference while compensating for the variations of theexternal perturbations. Indeed, using the bilinear system model, a statefeedback control has been developed resorting to Lyapunov stabilitytheory to stabilize the tracking error for nominal working conditions. Aphenomenological representation has been considered to describe thesystem behavior taking into account the uncertainty and the externaldisturbances in the plant. A robust control law can be established inorder to force the system dynamics to follow the nominal behavior usingthe phenomenological model.

Knowing that the nominal system is controlled using the inner loop totrack the predefined reference, the designed control will implicitlymaintain the system output around the desired temperature level.Simulation tests have proven that the presented control strategyachieves the control objective while ensuring robustness with respect toexternal disturbances, model uncertainties, and system aging without theneed to measure the external perturbations or to estimate theirefficient values. Convergence of the closed loop control has been provenusing the distributed parameters model of the ACUREX field.

Description of the Plant

A concentrated parabolic solar collector is a spatially distributedindustrial system working based on a heat transfer mechanism. Itincludes a set of parabolic mirrors concentrating the sunlight to heat afluid in a central pipe of the collector, which is routed to feed athermal process. Referring to FIG. 1A, shown is an image providing anoverall view of one line concentrated solar collector of a distributedsolar collector. FIG. 1B is a schematic diagram illustrating theradiation reflected on the concentrated solar collector.

Based on an energy analysis, while neglecting the heat losses and thethermal exchanges between the tube and the fluid, the heat transferalong the solar collector can be modeled by a hyperbolic partialdifferential equation where the fluid temperature is the system state:

$\begin{matrix}\left\{ \begin{matrix}{{{{T_{t}\left( {x,t} \right)} + {{u(t)}{T_{x}\left( {x,t} \right)}}} = {s\left( {x,t} \right)}},} \\{{{T\left( {0,t} \right)} = {T_{in}(t)}},} \\{{{T\left( {L,t} \right)} = {{T_{out}(t)} = {y(t)}}},}\end{matrix} \right. & (1)\end{matrix}$

for x∈ [0, L] to denote the position along the pipe and t∈

⁺ to represent the time. T(x, t) refers to the fluid temperature at acertain position x and time t, where:

${T_{t}\left( {x,t} \right)} \equiv {\frac{\partial{T\left( {x,t} \right)}}{\partial t}\mspace{20mu} {and}\mspace{20mu} {T_{x}\left( {x,t} \right)}} \equiv \frac{\partial{T\left( {x,t} \right)}}{\partial x}$

are the first derivatives with respect to time and space, respectively.The source term s(x, t) depends on the solar irradiance I(x, t) and thesystem control input u(t), which is a function of the fluid volumetricflow rate Q(t), can be expressed respectively as:

${s\left( {x,t} \right)} = {{\frac{v_{0}G}{\rho \; c\; A_{s}}\mspace{14mu} {and}\mspace{14mu} {u(t)}} = {\frac{Q(t)}{A_{s}}.}}$

Besides, T_(in) and T_(out) denote the measured boundary values of thefluid temperature, respectively, at the positions x=0 and x=L. Theremaining system parameters are summarized in the table of FIG. 2.

Bilinear Approximate Model of the Solar Collector

Model Approximation.

The procedure to approximate the first order hyperbolic distributedmodel of the solar collector by a low dimensional bilinear state spacerepresentation will now be discussed. A scheme to obtain numericalsolutions for partial differential equations (PDEs) is discussed in“Multiquarics—a scattered data approximation scheme with applications tocomputational fluid-dynamics” by E. J. Kansa (Computers Math. Applic.,vol. 19, no. 8/9, pp. 147-161, 1990), which is hereby incorporated byreference in its entirety. A dynamical version of this approach can beused to reproduce the temperature dynamics. Thereafter, the overallapproximate output at each position x along the collector can beinterpolated using radial basis functions defining the n sets D_(i) fori={1, . . . , n} using time varying weighting parameters ξ_(i)(t). Thus,the approximate value of the temperature {circumflex over (T)}(x, t) canbe given by:

$\begin{matrix}{{{\hat{T}\left( {x,t} \right)} = {{\sum\limits_{i = 1}^{n}{{\xi_{i}(t)}\; {\gamma_{i}(x)}}} = {{\gamma^{\top}(x)}{\xi (t)}}}},} & (2)\end{matrix}$

for i={1, . . . , n} with ξ(t) ∈

^(n) is the set of weighting parameters such that ξ(t)=[ξ₁(t), . . . ,ξ_(n)(t)]^(T) and γ(x)=[γ₁(x), . . . , γ_(n)(x)]^(T), where

$\begin{matrix}{{\gamma_{i}(x)} = {\frac{\mu_{i}(x)}{\sum\limits_{k = 1}^{n}{\mu_{k}(x)}}.}} & (3)\end{matrix}$

The radial basis functions μ_(i) characterize the n sets of D_(i), whichsubdivide the space [0, L], in terms of the spatial coordinates, suchthat:

D _(i)={(x,μ _(i)(x))|x∈[0,L]},i={1, . . . , n}.  (4)

The radial basis functions can be chosen to have Gaussian distributionscentered around n equidistant points along the spatial domain, wherethey are expressed in terms of the spatial positions as follows:

$\begin{matrix}{{{\mu_{i}(x)} = {\exp \left( {\frac{1}{2}\left( \frac{x - a_{i}}{\sigma_{i}} \right)^{2}} \right)}},} & (5)\end{matrix}$

where a_(i) and σ_(i) are the mean value and the standard deviation ofthe Gaussian function corresponding to the i^(th) set, such that fori={1, . . . , n}:

$\begin{matrix}{a_{i} = {{\left( {i - 1} \right)\frac{L}{\left( {n - 1} \right)}\mspace{14mu} {and}\mspace{20mu} \sigma_{i}} = {\sigma = {\frac{L}{\left( {n - 1} \right)}.}}}} & (6)\end{matrix}$

Referring to FIG. 3, shown is an example of the distribution of theradial basis functions along the collector.

Substituting the approximate solution {circumflex over (T)}(x, t) in thephysical distributed model of equation (1) leads to:

γ(x)^(T){dot over (ξ)}(t)+u(t)ξ^(T)(t)γ_(x)(x)=s(x,t),  (7)

and

{circumflex over (T)}(0,t)=γ^(T)(0)ξ(t)=T _(in)(t).  (8)

Thus, evaluating equation (7) at p different equidistant knotsconstituting a one dimensional grid in the domain of [0, L], oneobtains:

Γ{dot over (ξ)}(t)+Γ_(x)ξ(t)u(t)=S(t),  (9)

such that:

${\Gamma = \begin{bmatrix}{\gamma_{1}(0)} & \ldots & {\gamma_{n}(0)} \\{\gamma_{1}\left( {\Delta \; x} \right)} & \ldots & {\gamma_{n}\left( {\Delta \; x} \right)} \\\vdots & \; & \vdots \\{\gamma_{1}(j)} & \ldots & {\gamma_{n}(j)} \\\vdots & \; & \vdots \\{\gamma_{1}(L)} & \ldots & {\gamma_{n}(L)}\end{bmatrix}_{p \times n}},{\Gamma_{x} = \begin{bmatrix}{\gamma_{x\; 1}(0)} & \ldots & {\gamma_{xn}(0)} \\{\gamma_{x\; 1}\left( {\Delta \; x} \right)} & \ldots & {\gamma_{xn}\left( {\Delta \; x} \right)} \\\vdots & \; & \vdots \\{\gamma_{x\; 1}(j)} & \ldots & {\gamma_{xn}(j)} \\\vdots & \; & \vdots \\{\gamma_{x\; 1}(L)} & \ldots & {\gamma_{xn}(L)}\end{bmatrix}_{p \times n}},{{S(t)} = \begin{bmatrix}{s\left( {0,t} \right)} \\\vdots \\{s\left( {j,t} \right)} \\{s\left( {{j + {\Delta \; x}},t} \right)} \\\vdots \\{s\left( {L,t} \right)}\end{bmatrix}_{p \times \; 1}},{{{and}\mspace{14mu} {\gamma_{xi}(x)}} = \frac{\partial{\gamma_{i}(x)}}{\partial x}},{{{for}\mspace{14mu} j} = {{\left\{ {0,{\Delta \; x},{2\Delta \; x},\ldots \;,{\left( {p - 2} \right)\Delta \; x},L} \right\} \mspace{14mu} {with}\mspace{14mu} \Delta \; x} = {\frac{L}{p - 1}.}}}$

Subsequently, the time evolution of the parameters ξ(t) is given byminimization of the approximation error of the system of equation (9) ina least squares sense in order to ensure that the approximation isaccurately reproducing the distributed behavior of the temperature givenby equation (1). Therefore, the values of the parameters ξ(t) aredefined by the following control affine state space representation,which describes the dynamics of the parabolic solar collector, suchthat:

$\begin{matrix}\left\{ \begin{matrix}{{{\overset{.}{\xi}(t)} = {{A\; {\xi (t)}\mspace{20mu} {u(t)}} + {B(t)}}},} \\{{{y(t)} = {C\; {\xi (t)}}},}\end{matrix} \right. & (10)\end{matrix}$

where A_(n×n)=−(Γ^(T)Γ)⁻¹Γ^(T)Γ_(x) andB_(n×1)(t)=(Γ^(T)Γ)⁻¹Γ^(T)S(t)^(i).The system output representing the outlet fluid temperature is given by:

y(t)=Cξ(t)=T(L,t) with C _(1×n)=[γ₁(L) . . . γ_(n)(L)].

It is worth to point out that the system order is equal to the number ofinput sets n<p, which is generally less than ten (10). Moreover, thisstate dimension is relatively small compared to other numerical schemes,particularly the semi-discretization which usually uses a high statedimension in order to ensure the accuracy of the approximation.

The approximate model has been defined using a basis of eight Gaussianfunctions resulting in a state vector of dimension n=8 approximated on agrid of p=500 knots along the collector tube to construct the matrices Γand Γ_(x) of dimension 8×500. It has been shown that increasing thenumber of sets utilizes more computational effort without a significantimprovement in the performance. The standard deviation has been chosenequal to σ=L/(n−1). The procedure of the modified Gaussian interpolationapplied for approximation of the solar collector model can be enhancedwith a complete numerical analysis and performance evaluation.

Model Validation.

The bilinear model of equation (10) for the parabolic distributed solarcollector can be validated by comparison to the analytical solution ofthe physical model of equation (1). Knowing that (x, t) lies in theupper half-plane

²⁺:=(−∞, ∞)× [0,∞), the unknown is T:

²⁺→

and u:

⁺→

is a continuous function that is sufficiently smooth, the analyticalsolution of the hyperbolic partial differential equation describing theheat transfer dynamics along the solar collector takes the form of:

$\begin{matrix}\left\{ \begin{matrix}{{{{if}\mspace{14mu} x} - {\int_{0}^{t}{{u(\tau)}d\; \tau}}} \geq {0\text{:}}} \\{{{T\left( {x,t} \right)} = {{T\left( {{x - {\int_{0}^{t}{{u(\tau)}d\; \tau}}},0} \right)} + {\int_{0}^{t}{{s(\tau)}d\; \tau}}}},} \\{{{{else}\mspace{14mu} {if}\mspace{14mu} x} - {\int_{0}^{t}{{u(\tau)}d\; \tau}}} < {0\text{:}}} \\{{{T\left( {x,t} \right)} = {{T\left( {0,t^{*}} \right)} + {\int_{t^{*}}^{t}{{s(\tau)}d\; \tau}}}},{{{where}\mspace{14mu} {\int_{t^{*}}^{t}{{u(\tau)}d\; \tau}}} = {x.}}}\end{matrix} \right. & (11)\end{matrix}$

The parameters of the ACUREX field Plataforma Solar de Almeria were usedto run a validation test of the reduced model of equation (10). TheACUREX field can be considered to be a typical test bed for theconcentrated distributed technology. The approximate model was definedusing a basis of eight Gaussian functions (n=8) and a grid of 500 knotsalong the collector tube to construct the matrices Γ and Γ_(x). The testwas carried out for two (2) hours under a varying irradiance profilecomprising favorable working conditions and abrupt changes in theradiation level. FIG. 4 illustrates the profile of the solar irradianceduring the test. The model was also tested for different transfervelocities. The fluid volumetric flow rate was changed between differentranges within admissible physical limits as illustrated by the profileof the fluid flow rate in FIG. 5. The time evolution of the temperaturewas computed at different positions along the collector (L/5, 2L/5,3L/5, 4L/5, L). The obtained results were compared to the correspondinganalytical solutions of the hyperbolic heat transport equation.

Referring now to FIG. 6, shown are examples of the temperature dynamics(evolution) along the collector. The generated numerical approximationscorrelate with the analytical solutions and the thermal dynamics alongthe collector are well reproduced using the bilinear reduced model. Inorder to evaluate the accuracy level of the bilinear model of equation(10), a graphical analysis of the approximation error was performed andthe results illustrated in FIG. 7. A histogram of the approximationerror at the outlet position is provided in FIG. 8. It can be observedthat the generated error lies in the interval ([−0.3,0.3]) which is anacceptable error comprised by the reduction of the computational effort.Moreover, for a (60%) percentile, the error density is concentrated inthe neighborhood of zero bounded by (±0.05). Therefore, it can beconcluded that the bilinear approximate model presents satisfactoryresults and proves its accuracy in approximating the behavior of thedistributed solar collector. Moreover, the low dimension of theresulting state representation makes it suitable for real timeimplementation for analysis and/or control purposes.

Robust Lyapunov Controller

A robust controller can be utilized to stabilize the closed loop systemforcing the field temperature output to track a predefined referencedespite the changes in the external disturbances without reliance on themeasurements. A reference tracking problem can be formulated as twoembedded error stabilization problems. The inner closed loop can bedesigned for the stabilization of the nominal tracking error resortingto Lyapunov control theory to maintain the outlet temperature of thenominal system around a desired level. Nominal refers to the dynamics ofthe system for fixed working conditions, which can be used as areference for the system perturbations and parameters uncertainties. Theouter closed loop can be used to force the system output to track thenominal output using a phenomenological definition of the systemdynamics. By stabilizing the error dynamics between the system outputand the nominal output, the system output follows the output of thenominal system that is controlled to track the desired reference.Consequently, the control objective can be achieved.

When modeling concentrating solar power plants, the most common approachis to assume that all field equipment behaves similarly. Indeed, themodel presented in equation (1) has been used considering a homogeneoussource term along the solar field in order to simplify the controldesign. However, under external working conditions, the effective solarirradiance and/or the mirrors efficiency can be only locally evaluatedespecially for large production fields. Thus, the extrapolation of thelocal measurements is not a reasonable assumption given the naturalvariations. Indeed, scattered clouds may only affect the location of themeasurement position while the rest of the pipe may be under the effectof direct solar radiation, or vice versa. In addition, the mirrorsoptical efficiency can generally vary over time and can be inhomogeneousin space especially in a dusty humid environment. In order to design therobust controller, general working conditions of the system where thesource term is defined as a spatio-temporal function are considered.

Consequently, the source term can be defined by:

$\begin{matrix}{{{s_{g}\left( {x,t} \right)} = {\frac{{v_{0}\left( {x,t} \right)}G}{\rho \; {cA}_{s}}{I\left( {x,t} \right)}}},} & (12)\end{matrix}$

and can take the following form on the computational grid:

$\begin{matrix}{{{S_{g}(t)} = {{\frac{{v_{0}\left( {x,t} \right)}G}{\rho \; {cA}_{s}}{I\left( {x,t} \right)}} = \begin{bmatrix}{s\left( {0,t} \right)} \\\vdots \\{s\left( {k,t} \right)} \\{s\left( {{k + {\Delta \; x}},t} \right)} \\\vdots \\{s\left( {L,t} \right)}\end{bmatrix}_{p \times 1}}},} & (13)\end{matrix}$

for k={0, Δx, 2Δx . . . , L} and Δx=L/p−1. Therefore, the system can bemodeled by:

$\begin{matrix}\left\{ {\begin{matrix}{{{\overset{.}{\xi}(t)} = {{A\; {\xi (t)}\mspace{20mu} {u(t)}} + {B_{g}(t)}}},} \\{{{y(t)} = {C\; {\xi (t)}}},}\end{matrix}{with}} \right. & (14) \\{{B_{g}(t)} = {\left( {\Gamma^{\top}\Gamma} \right)^{- 1}\Gamma^{\top}{{S_{g}(t)}.}}} & (15)\end{matrix}$

A nominal representation for the state space model of equation (10) canbe defined as:

$\begin{matrix}\left\{ {\begin{matrix}{{{\overset{.}{\overset{\_}{\xi}}(t)} = {{A\; {\overset{\_}{\xi}(t)}\mspace{20mu} {\overset{\_}{u}(t)}} + \overset{\_}{B}}},} \\{{{\overset{\_}{y}(t)} = {C\overset{\_}{\; \xi}(t)}},}\end{matrix}{with}} \right. & (16) \\{{\overset{\_}{B} = {\left( {\Gamma^{\top}\Gamma} \right)^{- 1}\Gamma^{\top}\overset{\_}{S}}},} & (17)\end{matrix}$

such that:

$\begin{matrix}{\overset{\_}{S} = {\frac{\overset{\_}{v_{0}}G}{\rho \; {cA}_{s}}\overset{\_}{I\;}{1_{p \times 1}.}}} & (18)\end{matrix}$

where v₀ =1 and Ī=750 W/m². ξ and y denote the state and the output ofthe nominal model, respectively, describing the dynamics of the systemunder the nominal conditions of equation (18), and the unit vector ofdimension (p×1) is denoted by 1_(p×1).

Proposition 1.

Consider a system governed by the state representation of equation (10)and subjected to the unknown bounded disturbances S(t). The system ischaracterized by a nominal representation defined by equation (16) and aphenomenological model given by:

{dot over (y)}(t)=F(t)+αu(t),  (19)

where

-   -   a ∈        is a “non-physical” constant parameter for scaling; and    -   F is the compensation term, which carries the unknown and/or        nonlinear dynamics of the system as well as the varying external        disturbances.

Proof.

Proposition 1 introduces a phenomenological model representing thesystem dynamics with a linear expression of first order. Thisrepresentation approximates correctly the heat transport since the statespace representation of equation (10) is of relative degree one. Indeed,by computing the Lie derivative of the output vector field Cξ(t) withrespect to the input vector field Aξ(t), the following can be obtained:

${{\frac{\partial}{\partial\xi}\left( {C\; {\xi (t)}} \right)A\; {\xi (t)}} = {{C\; A\; {\xi (t)}} \neq 0}},$

which yields the relative degree of the system r=1. Therefore, thesystem output can be approximated by:

$\overset{.}{y} = {{\frac{\partial\;}{\partial\xi}{\left( {C\; {\xi (t)}} \right)\left\lbrack {{B(t)} + {A\; {\xi (t)}{u(t)}}} \right\rbrack}} = {{{CB}\; (t)} + {{CA}\; {\xi (t)}{{u(t)}.}}}}$

From this output {dot over (y)}, the system dynamics can be describedby:

{dot over (y)}(t)=F(t)+αu(t),  (19)

where α=CAξ(t)|_(ξ) ₀ and the additive term F(t) stands to represent thelinearized dynamics, the external disturbances and the linearizationuncertainties. Therefore, Proposition 1 holds for the system of equation(10).

The objective of the inner loop is to force the output of the nominalsystem to track the desired reference. The tracking reference problem ofthe nominal system is reduced to a study of the stabilizability of thetracking error where the system external disturbances are assumed to beconstant and equal to the nominal values as defined previously. Theissue under consideration is that of finding a control Lyapunov function(CLF) for the stabilization of the tracking error around the origin.Consequently, the corresponding feedback control will force the systemnominal output to track the desired reference.

Proposition 2.

Consider a system governed by the state representation of equation (16).Given a continuous state feedback control:

$\begin{matrix}{{{\overset{\_}{u}(t)} = \frac{{{- K}\; {\overset{\_}{e}(t)}} + {{\overset{.}{y}}_{r}(t)} - {C\; \overset{\_}{B}}}{C\; A\; {\overset{\_}{\xi}(t)}}},} & (20)\end{matrix}$

the nominal tracking error can be defined by:

ē(t)=C ξ(t)−y _(r)(t),  (21)

exhibits asymptotic convergence to the origin such that:

$\begin{matrix}{{{\lim\limits_{t\rightarrow\infty}{{\overset{\_}{e}(t)}}} = 0},} & (22)\end{matrix}$

by considering the following Lyapunov function:

$\begin{matrix}{{V:\left. {\mathbb{R}}\rightarrow{\mathbb{R}}^{+} \right.}{{{V\left( \overset{\_}{e} \right)} = {{\frac{1}{2}{\overset{\_}{e}}^{\top}\overset{\_}{e}} = {\left( {{C\overset{\_}{\; \xi}} - y_{r}} \right)^{\top}\left( {{C\; \overset{\_}{\xi}} - y_{r}} \right)}}},}} & (23)\end{matrix}$

where K ∈

⁺ is a positive constant.

Proof.

To prove that Proposition 2 holds for the system of equation (16),consider the tracking error defined in equation (21). The error dynamicscan be expressed as:

ė=C{dot over (ξ)}−{dot over (y)} _(r) =C[A ξ(t)u(t)+ B ]−{dot over (y)}_(r).  (24)

Resorting to Lyapunov nonlinear theory with the Lyapunov function ofequation (23), the stability of the unique equilibrium point ē=0 of theerror dynamics of equation (24) is analyzed. From (23), the derivativeof the Lyapunov function is expressed as:

{dot over (V)}=ė(t)ē(t)=(CA ξ(t)ū(t)+CB−{dot over (y)}_(r)(t))ē(t).  (25)

The dynamics of the error ē around its equilibrium point are stabilizedwhen the derivative function {dot over (V)} is negative definite. Theobjective is to find the adequate control input ū(t) within the possiblecontrol inputs set to satisfy this condition.

Without loss of generality, impose that {dot over (V)} takes the form:

{dot over (V)}=−Kē(t)ē(t)  (26)

which is negative definite as K∈

⁺ is a positive constant. K is a parameter to be tuned to control thetransient behavior of the system while respecting the physicallimitations of the system. After that, the control input can be deducedsuch that:

$\begin{matrix}{\overset{.}{V} = {{\left( {{{CA}\; \overset{\_}{\xi}\; (t){\overset{\_}{u}(t)}} + {C\; \overset{\_}{B}} - {{\overset{.}{y}}_{r}(t)}} \right){\overset{\_}{e}(t)}} = {{- \overset{\_}{K}}\; {\overset{\_}{e}(t)}{\overset{\_}{e}(t)}}}} & (27) \\{{Hence},{{\overset{\_}{u}(t)} = {\frac{{{- \overset{\_}{K}}\; {\overset{\_}{e}(t)}} - {C\; \overset{\_}{B}} + {{\overset{.}{y}}_{r}(t)}}{C\; A\overset{\_}{\; \xi}(t)}.}}} & (28)\end{matrix}$

Substituting the control law of equation (28) in equation (25), thederivative of the Lyapunov function of equation (23) is effectivelynegative definite. Therefore, Proposition 2 holds for the system ofequation (16).

Proposition 3.

Assume that Proposition 1 holds for the nonlinear model defined byequation (10). Then, the tracking error between the measured output andthe desired reference is stabilized by the control input given by:

$\begin{matrix}{{u(t)} = {{\frac{1}{\alpha}\left\lbrack {{- {F(t)}} + {C\; A\; {\overset{\_}{\xi}(t)}\; {\overset{\_}{u}(t)}} + {C\; \overset{\_}{B}} + {K\left( {{\overset{\_}{y}(t)} - {y(t)}} \right)}} \right\rbrack}.}} & (29)\end{matrix}$

where K∈

⁺ is a positive constant. Consequently, the tracking reference closedloop system presented in FIG. 9 exhibits an asymptotic referencetracking. The robust controller 900 includes an inner loop Lyapunovcontroller 903, an outer loop error stabilizer 906 and a nominal model909 of the process plant 912.

Proof.

Taking into consideration that Proposition 1 holds for the system ofequation (10) and Proposition 2 holds for its nominal representation ofequation (16), the stabilizability of the closed loop tracking error canbe defined by:

$\begin{matrix}\begin{matrix}{{e(t)} = {{y(t)} - {y_{r}(t)}}} \\{= {{y(t)} - {\overset{\_}{y}(t)} + {\overset{\_}{y}(t)} - {y_{r}(t)}}} \\{{= {{\underset{\_}{e}(t)} + {\overset{\_}{e}(t)}}},}\end{matrix} & (30)\end{matrix}$

where the difference between the nominal output and the real measuredsystem output is defined by the error e(t) such that:

e(t)= y (t)−y(t).

Consequently, the error dynamics under consideration are given by:

ė(t)=ė(t)+ė(t).  (31)

From Proposition 2, the error ē(t) is stabilized by ū(t) defined byequation (20). Therefore, to prove the convergence of the error e(t) inorder to ensure the stability of the global closed loop error e(t)defined by equation (30). The objective is to stabilize e(t) in order tohave the system output y(t) tracking y(t) and consequently tracking thereference y_(r)(t). Thus, the goal is to have:

ė(t)+ K e (t)={dot over (y)}(t)−{dot over (y)}(t)+ K ( y (t)−y(t))=0,  (32)

where K is a positive constant managing the rate of convergence of theerror. Without loss of generality, consider K=K=K. The output y(t) ismeasured and y(t) is computed using equation (16). Moreover,

{dot over (y)}(t)=CA ξ(t)ū(t)+CB,  (33)

and

{dot over (y)}(t)=CAξ(t)u(t)+CB(t).  (34)

However, the values of B(t) and ξ(t) are not available to define therelationship between {dot over (y)}(t) and the input u(t). To overcomethis problem, the phenomenological formulation defined in equation (19)can be used to describe the output variations in terms of the controlinput: Therefore, substituting equations (19) and (33) in equation (32)yields:

$\begin{matrix}{{u(t)} = {{\frac{1}{\alpha}\left\lbrack {{- {F(t)}} + {C\; A\; {\overset{\_}{\xi}(t)}\; {\overset{\_}{u}(t)}} + {C\; \overset{\_}{B}} + {K\left( {{\overset{\_}{y}(t)} - {y(t)}} \right)}} \right\rbrack}.}} & (35)\end{matrix}$

where ū(t) is defined by equation (20).

The control law of equation (35) depends explicitly on F(t), which iscontinuously updated based on equation (19) as follows:

F(t)={circumflex over ({dot over (y)})}(t)−αu(t−t _(w)),  (36)

The term {circumflex over ({dot over (y)})}(t) represents the estimatedvalue of the time derivative of the system output {dot over (y)}(t),which is numerically updated using a receding horizon algebraicderivative estimator, and t_(w) denotes the window width of the recedinghorizon strategy. The algebraic derivative estimation based on thereceding horizon approach has proven its efficiency in estimating thetime derivative of sampled, noisy time signals.

In this study, the first time derivative of the system output {dot over(y)}(t) was estimated considering a Taylor-series expansion of order N=2using the algebraic estimator, such that:

$\begin{matrix}{{{\overset{.}{\hat{y}}(t)} = {\int_{0}^{t_{w}}{\frac{{180\; \tau^{2}} - {192\; \tau \; t_{w}} + {36\; t_{w}^{2}}}{t_{w}^{4}}{y\left( {t - t_{w}} \right)}d\; \tau}}},} & (37)\end{matrix}$

Substituting equation (36) in equation (37), the control input is givenby:

$\begin{matrix}\begin{matrix}{{{\alpha \; {u(t)}} = {{- {F(t)}} + {C\; A\; {\overset{\_}{\xi}(t)}{\overset{\_}{u}(t)}} + {C\; B} + {K\left( {{\overset{\_}{y}(t)} - {y(t)}} \right)}}},} \\{{= {{- {\overset{.}{\hat{y}}(t)}} + {\alpha \; {u\left( {t - t_{w}} \right)}} - {K\; {\overset{\_}{e}(t)}} + {{\overset{.}{y}}_{r}(t)} + {K\left( {{\overset{\_}{y}(t)} - {y(t)}} \right)}}},} \\{= {{- {\overset{.}{\hat{y}}(t)}} + {\alpha \; {u\left( {t - t_{w}} \right)}} + {{\overset{.}{y}}_{r}(t)} + {{K\left( {{y_{r}(t)} - {y(t)}} \right)}.}}}\end{matrix} & (38)\end{matrix}$

Considering that the algebraic estimator reproduces correctly thederivative of the measured output and converges faster than the systemdynamics, it can be concluded that the estimation error is bounded suchthat:

|{dot over (y)}(t)−{circumflex over ({dot over (y)})}(t)|=|θ(t)|≤θ,θ∈

⁺.  (39)

Therefore, equation (38) can be rewritten as:

αu(t)=−{dot over (y)}(t)+αu(t−t _(w))+{dot over (y)} _(r)(t)+K(y_(r)(t)−y(t))+θ(t).  (40)

From equation (40), the tracking error dynamics are given by:

$\begin{matrix}{\begin{matrix}{{\overset{.}{e} + {K\; e}} = {{\alpha \left( {{u(t)} - {u\left( {t - t_{w}} \right)}} \right)} - {\theta (t)}}} \\{= {{\alpha \; \Delta \; {u(t)}} - {{\theta (t)}.}}}\end{matrix}{{Therefore},}} & (41) \\{{e(t)} = {{{\exp \left( {- {\int_{0}^{t}{K\; d\; \tau}}} \right)}\left\lbrack {{\int_{0}^{t}{{\exp \left( {\int_{0}^{\tau}{{Kd}\; \epsilon}} \right)}{{\alpha\Delta u}(\tau)}d\; \tau}} + {e(0)}} \right\rbrack}.}} & (42)\end{matrix}$

Knowing that K∈

⁺, Δu(t) and θ(t) are bounded and α is fixed, it can be concluded thatstarting from any initial conditions e(0),

$\begin{matrix}{{{\lim\limits_{t\rightarrow\infty}{{e(t)}}} = 0},} & (43)\end{matrix}$

i.e., the reference tracking is asymptotically achieved.

Evaluation Results

The parameters of the ACUREX field were considered for the numericaltests to evaluate the controller performance. The measured inlettemperature was provided as a boundary condition for the simulations.The time derivative of the sampled measured output using an algebraicestimator following equation (37), in order to update continuously F(t).Three different tests were carried out depending on the intensityprofile of the sunlight striking the solar collector along 5 hours. Thebehavior of the controller was first tested under ideally selectedcircumstances considering a smooth irradiance profile. The second testwas performed using a varying profile with some rapid changes in thedisturbances. The third test was carried out under unfavorable radiationconditions in order to evaluate the controller performance under extremeconditions. To take into consideration the mirror's cleanness factor andthe partial shading in the system, an inhomogeneous spatiallydistributed factor was multiplied by the time varying source term. FIG.10A graphically illustrates an example of an inhomogeneous distributionof the varying parameters. A variable Δ_(I)(x, t) was defined andpresented in FIG. 10B. The term Δ_(I)(x, t) is an inhomogeneousdistribution in time and space that defines the percentage ofinhomogeneity in space, such that:

$\begin{matrix}{{{s\left( {x,t} \right)} = {\frac{v\; G}{\rho \; c\; A_{s}}\left\lbrack {{I(t)} + {{\Delta_{I}\left( {x,t} \right)}{I(t)}}} \right\rbrack}},} & (44)\end{matrix}$

where I(t) denotes the time varying profiles of FIGS. 11A, 12A and 13A.It is worth noting that a hysteresis control has been applied on thegenerated control input in order to keep the fluid flow rate within theadmissible physical bounds. Indeed, lower fluid flow rates areundesirable in order to avoid the increase in the temperature of thethermal carrier fluid above the maximum recommended by the providerleading to its degradation.

The same configuration of the radial basis interpolation adoptedpreviously for the model validation was again considered for thesetests, where the generated outlet temperature is compared to the setpoint to evaluate the performance of the closed loop system of FIG. 9.FIGS. 11A-11C, 12A-12C and 13A-13C show examples of (a) solarirradiance, (b) reference tracking, and (c) control input for the first,second and third tests, respectively. Note that a hysteresis control hasbeen applied on the generated control input in order to keep the fluidflow rate within the admissible physical bounds. Indeed, lower fluidflow rates are undesirable in order to avoid increasing the temperatureof the thermal carrier fluid above the maximum recommended by theprovider leading to its degradation. Moreover, the control isconditioned by the physical limitations of the actuator, the pump.

Moreover, the control is conditioned by the physical limitations of theactuator (e.g., the pump). The response time of the reference trackingclosed loop can be managed by tuning the design parameter K (Lyapunovcontroller gain). Increasing the value of K makes the transient timefaster. For these series of tests, the gain was chosen to be K=0.005because different trials have shown that increasing this gain causesoscillations in the system response and might lead to instability anddivergence of the output.

Referring to FIGS. 11A-11C, shown are results of the first test. It canbe seen that the control objectives were achieved. Indeed, the robustcontrol strategy was able to cope with external perturbations and toforce the system outlet 1103 to follow the step variations in thereference 1106 without measuring or estimating the source termdistribution affecting the plant. Moreover, the generated control input1109 presented reasonable behavior and the changes with respect to thevariations in the temperature reference 1106 can be clearly observed.The flow rate was decreased when an increasing step reference wasinjected and it was raised (increased) for a decreasing set temperature.

Referring next to FIGS. 12A-12C, shown are results of the second test.The efficiency of the robust controller 900 of FIG. 9 was proven in thesecond test where the irradiance profile 1212 was affected by smallvariations. The closed loop stability was maintained using a controlinput signal 1209 within the admissible limits except for a short timearound 2 pm due to the low intensity of the sunlight. At that time thesolar irradiance 1212 striking the system suddenly decreased to zerowhich stopped the thermal production. Thereafter, it can be seen thatthe outlet temperature 1203 takes the value of the inlet temperature1215 because the external thermal source is not available.

Referring now to FIGS. 13A-13C, shown are results of the third test. Thesystem was tested under unfavorable circumstances with rapid variationsin the intensity 1312. As it can be seen at the beginning of the test,for a zero irradiance, the outlet temperature 1303 takes the value ofthe inlet temperature 1315 and the system does not produce thermalenergy. Furthermore, it can be observed that the controller of FIG. 9was efficient in stabilizing the closed loop system trying to keep theoutlet temperature 1303 around the desired reference value 1306 bycompensating for the external disturbances. However, due to the factthat the control input 1309 was constrained by physical limitations, itwas not possible to ensure the reference tracking for low intensitieswhere the fluid flow rate was reduced to a minimum to keep the outlettemperature 1303 as high as possible and consequently close to the setpoint (reference) 1306.

Note that the error converges to zero bounded by 5° C. despite thechanges in the working conditions without exact measurements orefficient estimation of these disturbances. In addition, the timeresponse was evaluated to be between 5 and 10 minutes for reference stepchanges ranging between 10° C. and 15° C., which is very promising inreal time implementation for thermal processes.

A robust controller 900 (FIG. 9) based on a low dimensional bilinearapproximate model of the parabolic distributed collector was presented.The model used for the control design has been derived from thehyperbolic transport equation as a result of a dynamical Gaussianinterpolation. It takes the form of a low dimensional state spacerepresentation reducing the computational effort and the designcomplexity while reproducing the heat transfer dynamics along the solarcollector with satisfactory accuracy.

An efficient controller was found in order to maintain the field outlettemperature around a certain desired level by tuning the fluid velocityin the collector tube despite environmental changes. The robust controlforces the trajectory of the system output to track a predefinedreference despite the unknown and unmeasured random variations of theexternal disturbances affecting the thermal dynamics. The controller hasbeen designed using a combination of a Lyapunov stabilizing statefeedback and a phenomenological representation of the system. Severalnumerical tests have been performed to evaluate the performance of thecontrol strategy under different levels and variations of the solarirradiance. The robust control performed well in terms of tracking errorstabilization and closed loop time response, in addition to therobustness with the respect to the external disturbances and theuncertainty in the system parameters.

With reference now to FIG. 14, shown is a schematic block diagram of anexample of processing circuitry 1400 that may be used to implementvarious portions of the robust controller 900 of FIG. 9 in accordancewith various embodiments of the present disclosure. The processingcircuitry 1400 includes at least one processor circuit, for example,having a processor 1403 and a memory 1406, both of which are coupled toa local interface 1409. To this end, the processing circuitry 1400 maybe implemented using one or more circuits, one or more microprocessors,microcontrollers, application specific integrated circuits, dedicatedhardware, digital signal processors, microcomputers, central processingunits, field programmable gate arrays, programmable logic devices, statemachines, or any combination thereof. The local interface 1409 maycomprise, for example, a data bus with an accompanying address/controlbus or other bus structure as can be appreciated. The processingcircuitry 1400 can include a display for rendering of generated graphicssuch as, e.g., a user interface and an input interface such, e.g., akeypad or touch screen to allow for user input. In addition, theprocessing circuitry 1400 can include communication interfaces (notshown) that allow the processing circuitry 1400 to communicativelycouple with other communication devices. The communication interfacesmay include one or more wireless connection(s) such as, e.g., Bluetoothor other radio frequency (RF) connection and/or one or more wiredconnection(s).

Stored in the memory 1406 are both data and several components that areexecutable by the processor 1403. In particular, stored in the memory1406 and executable by the processor 1403 are robust controlapplication(s) 1415, an operating system 1418, and/or other applications1421. Robust control applications 1415 can include applications thatsupport, e.g., controllers for control of the operation of uncertainsystems such as, e.g., solar collectors and/or observers for estimationof states and/or characteristics of the uncertain systems. It isunderstood that there may be other applications that are stored in thememory 1406 and are executable by the processor 1403 as can beappreciated. Where any component discussed herein is implemented in theform of software, any one of a number of programming languages may beemployed such as, for example, C, C++, C#, Objective C, Java®,JavaScript®, Perl, PHP, Visual Basic®, Python®, Ruby, Delphi®, Flash®,LabVIEW® or other programming languages.

A number of software components are stored in the memory 1406 and areexecutable by the processor 1403. In this respect, the term “executable”means a program file that is in a form that can ultimately be run by theprocessor 1403. Examples of executable programs may be, for example, acompiled program that can be translated into machine code in a formatthat can be loaded into a random access portion of the memory 1406 andrun by the processor 1403, source code that may be expressed in properformat such as object code that is capable of being loaded into a randomaccess portion of the memory 1406 and executed by the processor 1403, orsource code that may be interpreted by another executable program togenerate instructions in a random access portion of the memory 1406 tobe executed by the processor 1403, etc. An executable program may bestored in any portion or component of the memory 1406 including, forexample, random access memory (RAM), read-only memory (ROM), hard drive,solid-state drive, USB flash drive, memory card, optical disc such ascompact disc (CD) or digital versatile disc (DVD), floppy disk, magnetictape, or other memory components.

The memory 1406 is defined herein as including both volatile andnonvolatile memory and data storage components. Volatile components arethose that do not retain data values upon loss of power. Nonvolatilecomponents are those that retain data upon a loss of power. Thus, thememory 1406 may comprise, for example, random access memory (RAM),read-only memory (ROM), hard disk drives, solid-state drives, USB flashdrives, memory cards accessed via a memory card reader, floppy disksaccessed via an associated floppy disk drive, optical discs accessed viaan optical disc drive, magnetic tapes accessed via an appropriate tapedrive, and/or other memory components, or a combination of any two ormore of these memory components. In addition, the RAM may comprise, forexample, static random access memory (SRAM), dynamic random accessmemory (DRAM), or magnetic random access memory (MRAM) and other suchdevices. The ROM may comprise, for example, a programmable read-onlymemory (PROM), an erasable programmable read-only memory (EPROM), anelectrically erasable programmable read-only memory (EEPROM), or otherlike memory device.

Also, the processor 1403 may represent multiple processors 1403 and thememory 1406 may represent multiple memories 1406 that operate inparallel processing circuits, respectively. In such a case, the localinterface 1409 may be an appropriate network that facilitatescommunication between any two of the multiple processors 1403, betweenany processor 1403 and any of the memories 1406, or between any two ofthe memories 1406, etc. The local interface 1409 may comprise additionalsystems designed to coordinate this communication, including, forexample, performing load balancing. The processor 1403 may be ofelectrical or of some other available construction.

Although the robust control application(s) 1415, the operating system1418, application(s) 1421, and other various systems described hereinmay be embodied in software or code executed by general purpose hardwareas discussed above, as an alternative the same may also be embodied indedicated hardware or a combination of software/general purpose hardwareand dedicated hardware. If embodied in dedicated hardware, each can beimplemented as a circuit or state machine that employs any one of or acombination of a number of technologies. These technologies may include,but are not limited to, discrete logic circuits having logic gates forimplementing various logic functions upon an application of one or moredata signals, application specific integrated circuits havingappropriate logic gates, or other components, etc. Such technologies aregenerally well known by those skilled in the art and, consequently, arenot described in detail herein.

Also, any logic or application described herein, including the robustcontrol application(s) 1415 and/or application(s) 1421, that comprisessoftware or code can be embodied in any non-transitory computer-readablemedium for use by or in connection with an instruction execution systemsuch as, for example, a processor 1403 in a computer system or othersystem. In this sense, the logic may comprise, for example, statementsincluding instructions and declarations that can be fetched from thecomputer-readable medium and executed by the instruction executionsystem. In the context of the present disclosure, a “computer-readablemedium” can be any medium that can contain, store, or maintain the logicor application described herein for use by or in connection with theinstruction execution system. The computer-readable medium can compriseany one of many physical media such as, for example, magnetic, optical,or semiconductor media. More specific examples of a suitablecomputer-readable medium would include, but are not limited to, magnetictapes, magnetic floppy diskettes, magnetic hard drives, memory cards,solid-state drives, USB flash drives, or optical discs. Also, thecomputer-readable medium may be a random access memory (RAM) including,for example, static random access memory (SRAM) and dynamic randomaccess memory (DRAM), or magnetic random access memory (MRAM). Inaddition, the computer-readable medium may be a read-only memory (ROM),a programmable read-only memory (PROM), an erasable programmableread-only memory (EPROM), an electrically erasable programmableread-only memory (EEPROM), or other type of memory device.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations setforth for a clear understanding of the principles of the disclosure.Many variations and modifications may be made to the above-describedembodiment(s) without departing substantially from the spirit andprinciples of the disclosure. All such modifications and variations areintended to be included herein within the scope of this disclosure andprotected by the following claims.

It should be noted that ratios, concentrations, amounts, and othernumerical data may be expressed herein in a range format. It is to beunderstood that such a range format is used for convenience and brevity,and thus, should be interpreted in a flexible manner to include not onlythe numerical values explicitly recited as the limits of the range, butalso to include all the individual numerical values or sub-rangesencompassed within that range as if each numerical value and sub-rangeis explicitly recited. To illustrate, a concentration range of “about0.1% to about 5%” should be interpreted to include not only theexplicitly recited concentration of about 0.1 wt % to about 5 wt %, butalso include individual concentrations (e.g., 1%, 2%, 3%, and 4%) andthe sub-ranges (e.g., 0.5%, 1.1%, 2.2%, 3.3%, and 4.4%) within theindicated range. The term “about” can include traditional roundingaccording to significant figures of numerical values. In addition, thephrase “about ‘x’ to ‘y’” includes “about ‘x’ to about ‘y’”.

1. A system, comprising: a process plant; and a robust Lyapunovcontroller configured to control an input of the process plant, therobust Lyapunov controller comprising an inner closed loop Lyapunovcontroller and an outer closed loop error stabilizer.
 2. The system ofclaim 1, wherein the process plant is a distributed solar collector andthe input of the process plant is an inlet fluid flow rate.
 3. Thesystem of claim 1, wherein the robust Lyapunov controller comprises anominal model of the process plant configured to generate an estimatedoutput based at least in part upon fixed working conditions of theprocess plant.
 4. The system of claim 3, wherein the nominal model is aphysical distributed model of the process plant.
 5. The system of claim4, wherein the physical distributed model is a bilinear model thatapproximates the process plant by a low order nonlinear set of ordinarydifferential equations using dynamical Gaussian interpolation.
 6. Thesystem of claim 1, wherein the outer closed loop error stabilizer isconfigured to generate the input to force an output of the process plantto track a nominal output using a phenomenological representation of theprocess plant.
 7. The system of claim 8, wherein the inner closed loopLyapunov controller is configured to generate an estimated input basedupon an estimated output generated by a nominal model of the processplant and an output reference, the estimated input provided to the outerclosed loop error stabilizer and the nominal model of the process plant.8. The system of claim 1, wherein the process plant comprises aparabolic solar collector.
 9. A method, comprising: monitoring an outputof a process plant; generating, by an inner closed loop Lyapunovcontroller, an estimated control input based at least in part upon adefined output reference; generating, by an outer closed loop errorstabilizer, a control input based at least in part upon the estimatedcontrol input and a compensation term based upon a phenomenologicalmodel of the process plant; and adjusting operation of the process plantbased upon the control input to force the output of the process plant totrack the defined output reference.
 10. The method of claim 9, whereinthe compensation term is determined based upon the control input and arate of change of the output.
 11. The method of claim 9, wherein thecontrol input is further based upon a difference between an estimatedoutput and the output being monitored, where the estimated output isbased upon a nominal model of the process plant and the estimatedcontrol input.
 12. The method of claim 11, wherein the nominal model isa physical distributed model of the process plant.
 13. The system ofclaim 12, wherein the physical distributed model is a bilinear modelthat approximates the process plant by a low order nonlinear set ofordinary differential equations using dynamical Gaussian interpolation.14. The method of claim 11, wherein the control input is further basedupon time varying weighting parameters provided by the nominal model.15. The method of claim 9, wherein the process plant is a distributedsolar collector and the input of the process plant is an inlet fluidflow rate.
 16. A system of a process plant, comprising: processingcircuitry including a processor and a memory, and a local interfacecoupled to the processor and the memory, wherein one or more robustcontrol applications are stored in the memory and executable by theprocessor, wherein the robust control applications support a robustLyapunov controller configured to control an input of the process plant,wherein the robust Lyapunov controller includes a nominal model of aprocess plant, an inner closed loop Lyapunov controller, and an outerclosed loop error stabilizer.
 17. The system of claim 16, wherein thenominal model of the process plant is configured to generate anestimated output based at least in part upon fixed working conditions ofthe process plant.
 18. The system of claim 17, wherein the nominal modelis a physical distributed model of the process plant.
 19. The system ofclaim 16, wherein the outer closed loop error stabilizer is configuredto generate the input to force an output of the process plant to track anominal output using a phenomenological representation of the processplant.
 20. The system of claim 16, wherein the inner closed loopLyapunov controller is configured to generate an estimated input basedupon an estimated output generated by a nominal model of the processplant and an output reference, the estimated input provided to the outerclosed loop error stabilizer and the nominal model of the process plant.